Some topological aspects of ideals in quantales
Submitted: 2024-09-30
|Accepted: 2025-01-15
|Published: 2025-04-01
Copyright (c) 2025 Amartya Goswami
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
Quantale, quasi-compactness, sobriety, spectral spaces
Supporting agencies:
Abstract:
As a natural extension of the ongoing development of a theory of ideals in commutative quantales with an identity element, this article aims to study into the analysis of certain topological properties exhibited by distinguished classes of ideals. These ideals are equipped with quantale topologies. The primary objectives encompass characterizing quantale spaces exhibiting sobriety, examining several conditions pertaining to quasi-compactness, and demonstrating that quantale spaces comprised of proper ideals adhere to the spectral properties as defined by Hochster. We introduce the notion of a strongly disconnected spaces and show that for a quantale with zero Jacobson radical, strongly disconnected quantale spaces containing all maximal ideals of the quantale imply existence of non-trivial idempotent elements in the quantale. Additionally, a sufficient criterion for establishing the connectedness of a quantale space is presented. Finally, we discuss on continuous maps between quantale spaces.
References:
J. Y. Abuhlail, A Zariski topology for modules, Commun. Algebra 39, no. 11 (2011), 4163-4182. https://doi.org/10.1080/00927872.2010.519748
J. Y. Abuhlail, Zariski topologies for coprime and second submodules, Algebra Colloq. 22 (2015), 47-72. https://doi.org/10.1142/S1005386715000061
R. Ameri, Some properties of Zariski topology of multiplication modules, Houst. J. Math. 36, no. 2 (2010), 337-344.
D. D. Anderson, Multiplicative lattices, PhD dissertation, University of Chicago, 1974.
A. Azizi, Strongly irreducible ideals, J. Aust. Math. Soc. 84 (2008), 145-154. https://doi.org/10.1017/S1446788708000062
H. Ansari-Toroghy and F. Farshadifar, The Zariski topology on the second spectrum of a module, Algebra Colloq. 21, no. 4 (2014), 671-688. https://doi.org/10.1142/S1005386714000625
J. Abuhlail and H. Hroub, Zariski-like topologies for lattices with applications to modules over associative rings, J. Alg. Appl. 18, no. 7 (2019), 1950131. https://doi.org/10.1142/S0219498819501317
H. Ansari-Toroghy and S. S. Pourmortazavi, On the minimal submodules of a module, Matematiche (Catania) 72, no. 1 (2017), 185-200.
H. Ansari-Toroghy, S. Keyvani, and F. Farshadifar, The Zariski topology on the second spectrum of a module (II), Bull. Malays. Math. Sci. Soc. 39, no. 3 (2016), 1089-1103. https://doi.org/10.1007/s40840-015-0225-y
N. Bourbaki, Elements of mathematics: Commutative Algebra, Addison-Wesley, Reading, MA, 1972.
M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra 4 (2008), 104-130.
R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481-498. https://doi.org/10.2140/pjm.1962.12.481
R. P. Dilworth, Abstract residuation over lattices, Bull. Amer. Math. Soc. 44 (1938), 262-268. https://doi.org/10.1090/S0002-9904-1938-06736-5
T. Dube and A. Goswami, Ideal spaces: An extension of structure spaces of rings, J. Algebra Appl. 22, no. 11 (2023), 2350245. https://doi.org/10.1142/S0219498823502456
M. Dickmann, N. Schwartz, and M. Tressel, Spectral spaces, Cambridge Univ. Press, 2019. https://doi.org/10.1017/9781316543870
A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices, in Algebra and coding theory, 113-131, Contemp. Math. 785, Amer. Math. Soc., 2023. https://doi.org/10.1090/conm/785/15780
A. Facchini, C. A. Finocchiaro, and G. Janelidze, Abstractly constructed prime spectra, Algebra Universalis 83(8) (2022), 38 pp. https://doi.org/10.1007/s00012-021-00764-z
A. Facchini, M. Fontana, and D. Spirito, A topological version of Hilbert's Nullstellensatz, J. Algebra 461 (2016), 25-41. https://doi.org/10.1016/j.jalgebra.2016.04.020
A. Facchini, A. Goswami, and D. Spirito, Distinguished classes of ideal spaces and their topological properties, Comm. Algebra 51, no. 4 (2023), 1752-1760. https://doi.org/10.1080/00927872.2022.2141767
A. Facchini and D. Spirito, Suprema in spectral spaces and the constructible closure, New York J. Math. 26 (2020), 1064-1092.
G. Georgescu, Flat topology on the spectra of quantales, Fuzzy Sets and Systems 406 (2021), 22-41. https://doi.org/10.1016/j.fss.2020.08.009
J. S. Golan, Semirings and their applications, Springer, 1999. https://doi.org/10.1007/978-94-015-9333-5
A. Goswami, Proper spaces are spectral, Appl. Gen. Topol. 24, no. 1 (2023), 95-99. https://doi.org/10.4995/agt.2023.17800
A. Goswami, On ideals in quantales, I, arXiv:5045264 [math.RA].
A. Goswami, Primitive ideals and Jacobson's structure spaces of semigroups, Discuss. Math. Gen. Algebra Appl., to appear.
A. Grothendieck, Éléments de géométrie algébrique I, Le langage des schémas, Inst. Hautes Études Sci. Publ. Math., No. 4, 1960. https://doi.org/10.1007/BF02684778
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous lattices and domains, Cambridge Univ. Press, 2003. https://doi.org/10.1017/CBO9780511542725
M. Hochster, Prime ideal structure in commutative rings, Trans. Am. Math. Soc. 142 (1969), 43-60. https://doi.org/10.1090/S0002-9947-1969-0251026-X
M. Henriksen and M. Jerison, The space of minimal prime ideals in a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. https://doi.org/10.1090/S0002-9947-1965-0194880-9
K. Hila, S. Kar, Sh. Kuka, and K. Naka, Some structure spaces of generalized semirings, Filomat 32, no. 13 (2018), 4461-4472. https://doi.org/10.2298/FIL1813461H
K. Iseki, Ideal theory of semiring, Proc. Japan Acad. 32 (1956), 554-559. https://doi.org/10.2183/pjab1945.32.201
K. Iseki, Notes on topological spaces. V. On structure spaces of semiring, Proc. Japan Acad. 32 (1956), 426-429. https://doi.org/10.2183/pjab1945.32.201
K. Iseki and Y. Miyanaga, Notes on topological spaces. III. On space of maximal ideals of semiring, Proc. Japan Acad. 32 (1956), 325-328. https://doi.org/10.3792/pja/1195525375
N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring, Proc. Nat. Acad. Sci. U.S.A. 31 (1945), 333-338. https://doi.org/10.1073/pnas.31.10.333
D. Kruml and J. Paseka, Algebraic and categorical aspects of quantales, in Handbook of algebra, vol. 5, Elsevier, 2008. https://doi.org/10.1016/S1570-7954(07)05006-1
N. H. McCoy, Rings and ideals, Mathematical Association of America, 1948. https://doi.org/10.5948/9781614440086
J. D. McKnight Jr., On the characterization of rings of functions, Purdue doctoral thesis, 1953.
C. J. Mulvey, &, Rend. Circ. Mat. Palermo (2) Suppl. 12 (1986), 99-104.
R. L. McCasland and P. F. Smith, Zariski spaces of modules over arbitrary rings, Comm. Algebra 34, no. 11 (2006), 3961-3973. https://doi.org/10.1080/00927870600875880
R. L. McCasland, M. E. Moore, and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25 (1997), 79-103. https://doi.org/10.1080/00927879708825840
B. Olberding, Topological aspects of irredundant intersections of ideals and valuation rings, in Multiplicative ideal theory and factorization theory 170 (2016), 277-307. https://doi.org/10.1007/978-3-319-38855-7_12
H. A. Priestley, Intrinsic spectral topologies, in Papers on general topology and applications (Flushing, NY, 1992), 728, 78-95, New York Acad. Sci., New York, 1994. https://doi.org/10.1111/j.1749-6632.1994.tb44135.x
K. I. Rosenthal, Quantales and their applications, Pitman research notes in Math. No. 234, Longman, Scientific and Technical, 1990.
M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481. https://doi.org/10.1090/S0002-9947-1937-1501905-7
U. Tekir, The Zariski topology on the prime spectrum of a module over noncommutative rings, Algebra Colloq. 16, no. 4 (2009), 691-698. https://doi.org/10.1142/S1005386709000650
G. Ulucak, U. Tekir, and K. P. Shum, On Spec(M) and separation axioms between T0 and T1, Southeast Asian Bull. Math. 39 (2015), 717-725.
M. Ward, Residuation in structures over which a multiplication is defined, Duke Math. J. 3 (1937), 627-636. https://doi.org/10.1215/S0012-7094-37-00351-X
M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335-354. https://doi.org/10.1090/S0002-9947-1939-1501995-3
